Solution: 2015 Fall Final - 11

Author: Michiel Smid

Question

Nick gets 75 bananas for his birthday. He decides to eat them all over a period of 5 days. In order to do this, Nick makes a banana-schedule, which specifies the number of bananas he is going to eat on the first day, on the second day, etc., up to the fifth day. For example, (20, 20, 10, 20, 5), (40, 13, 0, 20, 2), and (40, 13, 20, 2, 0) are three different banana-schedules. What is the total number of banana-schedules?

(a)

${79 \choose 4}$

(b)

${80 \choose 4}$

(c)

${79 \choose 5}$

(d)

${80 \choose 5}$

Solution

Let’s use dividers to separate the bananas eaten per day.

We have 75 bananas and 4 dividers.

$x_1$ represents the number of bananas to the left of the first divider
$x_2$ represents the number of bananas between the first and second dividers
$x_3$ represents the number of bananas between the second and third dividers
$x_4$ represents the number of bananas between the third and fourth dividers
$x_5$ represents the number of bananas to the right of the fourth divider

Since we have added positions for dividers to the 75 bananas, we have 79 positions in total.

We need to choose 4 positions for the dividers: $ \binom{79}{4} $